A166600 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 19, 342, 6156, 110808, 1994544, 35901792, 646232256, 11632180608, 209379250944, 3768826516992, 67838877305856, 1221099791505237, 21979796247091188, 395636332447586151, 7121453984055556524

Offset

0,2

Comments

The initial terms coincide with those of A170738, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (17,17,17,17,17,17,17,17,17,17,17,-153).

Formula

G.f.: (t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(153*t^12 - 17*t^11 - 17*t^10 - 17*t^9 -17*t^8 -17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 -17*t + 1).

From G. C. Greubel, Dec 08 2024: (Start)

a(n) = 17*Sum_{j=1..11} a(n-j) - 153*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 18*x + 170*x^12 - 153*x^13). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-18*t+170*t^12-153*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 18 2016; Dec 08 2024 *)
coxG[{12, 153, -17}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 05 2016 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^12)/(1 - 18*x+170*x^12-153*x^13) )); // G. C. Greubel, Dec 08 2024
(SageMath)
def A166600_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-18*x+170*x^12-153*x^13) ).list()
print(A166600_list(40)) # G. C. Greubel, Dec 08 2024

Crossrefs

Cf. A154638, A169452, A170738.

Sequence in context: A165341 A165881 A166413 * A167049 A167126 A167676

Adjacent sequences: A166597 A166598 A166599 * A166601 A166602 A166603

Keyword

nonn

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Status

approved